A LINK BETWEEN TWO CHARACTERISATIONS OF COMPLETE MATRIX RINGS

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1 BULL. AUSTRAL. MATH. SOC. VOL. 51 (1995) [ ] 16S50, 16D70 A LINK BETWEEN TWO CHARACTERISATIONS OF COMPLETE MATRIX RINGS N.J. GROENEWALD AND L. VAN WYK We establish a link between the two characterizations, independently obtained in 1991 by Fuchs and Robson, of complete matrix rings in terms of the existence of nilpotent elements. Every ring herein is associative with identity, and matrix ring will mean complete matrix ring of size n ^ 2. Apart from the well known characterisations of matrix rings, Fuchs [1] and Robson [3] independently gave in 1991 new criteria for a ring to be a matrix ring. Following a characterisation of matrix rings of size 2 in [2, Theorem III.2], Fuchs showed in [1, Theorem 1] that a ring R is isomorphic to a matrix ring M,,(5), for some ring S, if and only if there are elements x and y in R such that (1) «""' ^ 0, x" = 0 = y 2, x+yis invertible and Annfx"" 1 } D Ry = {0}, where Ann{a; n ~ 1 } denotes the left annihilator of x n ~ 1. We first note that the condition a;"" 1 ^ 0 in (1) is superfluous, since if x n ~ 1 = 0, then y = 0, otherwise Ann{x n ~ 1 }nry contains the nonzero element y. But then x is invertible, contradicting the nilpotency of x. Robson proved in [3, Theorem 2.2] that R = Mn(S), for some ring S, if and only if there are elements x and y in R such that x n = 0 and (2) x n - x y + x n ~ 2 yx + + xyx n ~ 2 + yx 71 ' 1 = 1. Robson's method of proof involved the characterisation of a matrix ring M,,(5) as a direct sum of n mutually isomorphic right ideals, whereas Fuchs invoked the characterisation of a matrix ring in terms of a set {eij : 1 ^ i, j ^ n} of matrix units, that is, n Y^, en = 1 and eijeu 6jk e ih $jk being the Kronecker delta. i=i This short sequel was inspired by the observation that x + y is its own inverse in the proof of (i) =* (ii) in [2, Theorem III.2]. Consider now the following criterion, Received 10th August, 1994 Copyright Clearance Centre, Inc. Serial-fee code: /95 SA

2 512 N.J. Groenewald and L. van Wyk [2] which is a combination of (2) and some of the conditions in (1): there are elements x and y in a ring R such that (3) x n = 0=y 2, x n - 1 y+x n - 2 yx+-+xyx n - 2 +yx n - 1 = 1 and Annjx"- 1 } D Ry = {0}. We first show in Theorem 1 that (3) explicitly yields, in terms of x and y, on the one hand, the inverse of x + y, and on the other hand, a set of matrix units. This drastically simplifies the proof by Fuchs that the mentioned set is indeed a set of matrix units. Second, we prove in Proposition 2 that the intersection condition in (3) follows from the other conditions in (3) if n 2 or 3, but we show in Example 3 that this is not necessarily so if n ^ 4. Fuchs observed in [1, Corollary 4] that the intersection condition in (1) follows from the other conditions in (1) if n = 2. Example 3 also shows that the intersection condition in (1) does not necessarily follow from the other conditions in (1) if n ^ 4. Finally, Example 4 shows that the intersection condition in (1) does not necessarily follow from the other conditions in (1) if n = 3. Before we formally state the first result, we assume that x and y are elements of a ring R such that x n 0 = y 2 and such that (2) holds, and we point out three equalities. If we premultiply (2) by y, then (4) yx n ~ 1 y + yx n ~ 2 yx + + yxyx n ~ 2 = y, since y 2 = 0. Similarly, postmultiplication of (2) by y shows that x n ~ 2 yxy + + xyx n ~ 2 y + yx n ~ 1 y = y, and so by (4) (5) yx n ~ 2 yx + h yxyx n ~ 2 = x n ~ 2 yxy + + xyx n ~ 2 y. Next, if we simultaneously premultiply (2) by yx n ~ 2 and postmultiply (2) by y, then (6) yx n - 2 (xyx n - 2 )y + yx n ~ 2 (yx^y = yx n ~ 2 y, since x n = 0. THEOREM 1. If a. ring R contains elements x and y such that x n = 0 = y 2, x n ~ 1 y + x n ~ 2 yx + + xyx n ~ 2 + yx n ~ x = 1 and Ann{z n ~ 1 } C\ Ry = {0} for some n ^ 2, then (a; + y)" 1 = x n-1 + x n ~ 2 y + x n ~*yx H h xyx n ~ 3 + yx n ~ 2 and {x x ~ l yx n ~i : 1 < i,j < n} is a set of matrix units in R (which implies that R is a matrix ring). PROOF: We first show that (7) yx n - 1 y = y,

3 [3] Complete matrix rings 513 and, if n ^ 3, that (8) yx i y = 0, j = l,...,n-2. It follows directly from (2) (or (4)) that yxy = y in case n = 2. If n ^ 3, then by (5) we have (yx n ~ 2 y + + yxyx n ~ 3 )x = (x n ~ 2 yx -\ + xyx n ~ 2 )y Ann^"" 1 } n Ry = {0}. Therefore (4) implies (7). We conclude from (6) and (7) that yx n ~ 2 y + yx n ~ 2 y = yx n ~ 2 y, that is, yx n ~ 2 y = 0, and so (8) holds in case n 3. If n ^ 4, then let 2 < fc ^ n 1, and suppose now that yx n ~*y 0 for i = 2,...,fc 1. Then, again since x n = 0, simultaneous premultiplication of (2) by yx n ~ k and postmultiplication of (2) by y shows that yx n ~ k {x k ~ 1 yx n ~ k )y + yx n ~ k (x k ~ 2 yx n - k+1 )y + h yx n ~ k (xyx n - 2 )y + yx n ~ k {yx^^y = (yx n - 1 y)x n - k y + (yx n - 2 y)x n - ( - k - 1 '*y + + (yx n ~( k ~ 1 )y)x n ~ 2 y + yx n ~ k (yx n ~ 1 )y = yx n ~ k y. Hence, by (7) and the induction hypothesis we have yx n ~ k y + yx n ~ k y = yx n ~ k y, and so yx n ~ k y 0. Therefore induction establishes (8). For 1 ^ i,j ^ n, we set e<j := x l ~ 1 yx n ~'. Let 1 ^ k,l ^ n. We conclude from (7) that e^tji = x i ~ 1 yx n ~ 1 yx n ~ l -en. If j ^ fc, then (8) and the fact that x n = 0 = y 2 imply that e <; ejtj = x i ~ 1 yx n -' +k - 1 yx n ~ l =0. By (2) n we have ^ en = 1, and so {eij : 1 ^ t,j ^ n} is a set of matrix units. Fit=i nally, if one invokes (2), (8) and the nilpotency of x and y, then direct verification shows that (x + y)(x n ~ 1 + x n ~ 2 y + x n ~ 3 yx + + xyx n ~ s + yx n ~ 2 ) = 1 = (x"- 1 +x n - 2 y + x n ~ 3 yx xyx n ~ 3 +yx n ~ 2 )(x+y). D PROPOSITION 2. Let n 2 or 3. If R is a ring containing elements x and y such that x n - 0 = y 2 and x n ~ x y + x n ~ 2 yx + + xyx n ~ 2 + yx 71 ' 1 = 1, tien PROOF: If n = 2 and ry G Ann{z} n Ry, r G R, then ry = ry(xy +yx) = 0. Next, let n = 3. Then (5) states that (9) yxyx = xyxy. Also, if we premultiply (2) by yxy, then yxyx 2 y + yxyxyx yxy, and so by (9) (I) yxyx 2 y = yxy. Similarly, postmultiplication of (2) by yxy shows that (II) yx 2 yxy = yxy. But (6) states that yx 2 yxy + yxyx 2 y = yxy, and so by (10) and (11) we have (12) yxy = 0.

4 514 N.J. Groenewald and L. van Wyk [4] Consequently, if ry G Ann {a; 2 } n Ry, r R, then ry = ry(x 2 y + xyx + yx 2 ) = (ryx 2 )y + r(yxy)x +ry 2 x 2 =0. D We now exhibit, for every n ^ 4, a ring R(n) containing elements x and y such that a;"" 1 ^ 0, x n = 0 = y 2, x + y is invertible and x n ~ 1 y + x n ~ 2 yx H (- xyx n ~ 2 + yx 71-1 = 1, but Ann{aj n - 1 } n #(71)1/ ^ {0}, which shows, firstly, that the intersection condition does not necessarily follow from the other conditions in (3) if n ^ 4, and, secondly, that the intersection condition neither necessarily follows from the other conditions in (1) if n ^ 4. EXAMPLE 3. Let 5 be any ring and set R{n) := M,,(5), n ~ 4. We use the standard notation Eij for the matrix in M«(5) with I5 (the identity of 5) in position (i,j) and zeros elsewhere. Set x := E 2,i + ^3,2 -\ h -^n.n-i. 4,1 and y := ^i in + E 2lTl -i. Then -Sfc+1,1 + Ek+2,2 H +-^71,71-* Ek+3,1, if 1 ^ fc ^ n 3; t E-n-i.i + -Sn,2, if A = n 2; (13) x* = I Hence 0, if A; = n. I J.rrJ X y ^= JUfi 7i) T/»C ^ -*^1 1) Xlc/* I == JBli!/i 2 T~ -^2 1/ ~~ *- J 2 2~i *-'3 1 -*^4 2) and -1^3,3 ~r -^4,2 -^3,1) II 71 4, 4,4+ 5,3-5,1, ifn = 5; En-l,n-l+^n,n-2, if n > 6. If n ^ 5 and 2 <A: ^ n - 3, then 2<n-l-Jfc^n-3, and so by (13),n-l Ek+3,n)(E n -k,l + -En-fc+1,2 H +,t Ek+2,n-lE n -k+2,l (16) ^3,3 -^3,1 + Ei >2 5,3 + 5,1, if k = 2; = { 4,4 + 5,3-5,1 - #6,4, if n ^ 6 andfc = 3;

5 [5] Complete matrix rings 515 We conclude from (13)-(16) that x n ~ 1 y + x n ~ 2 yx + + xyx n ~ 2 + yz"" 1 4,4 + (-#3,3 + 4,2 - E 3A ) + (E 2, 2 + 3,1-4,2) + JEi,i, if n = 4; #5,5 + (-#4,4 + J#5,3 -#5,1) + (-#3,3 3,1 + -#4,2-5,3 + -#5,1) + (#2,2 + 3,1-4,2) + 1,1, if n = 5; #6,6 + (-#5,5 + E 6,i) + (-#4,4 + -#5,3 -#5,1 -#6,4) + (-#3,3 #3,1 + -#4,2 -#5,3 + -#5,l) + (-#2,2 + -#3,1 -#4,2) + -#1,1, if n = 6; n-3 En,n + (-#n-l,n-l + #71,11-2) + X) (-#ib+l,ib+l + -#*+2,* -#ib+3,fc+l) Jfc=4 + (-#4,4 + -#5,3 -#5,1 -#6,4) + (-#3,3 -#3,1 + -#4,2 -#5,3 + -#5,l) = 1,1 -f E 2,2 + + E n<n = l R(n) for every n ^ 4. However, Ann^"" 1 } D R(n)y = S.#i, n -i + 5E 2, n _i + + 5E n, n _i ^ {0}. Also note that #1,2 -#l,n + -#2,3 + -#3,2 + -#3,4 -#3,n +E 4,5 + "- + ^n-i,n + ^»,i, ifn^5; \{E X>2 - Ei A ) + E 2t3 + (-#3,2 + -#3,4) + -#4,i, if 2 is invertible in 5 and n = 4. Finally we construct a ring.r(3) containing elements x and y such that x 2 0, x 3 = 0 - y 2 and x + y is invertible, but Ann{a; 2 } 0 R{3)y ^ {0}. EXAMPLE 4. Here Z denotes the ring of integers and Z4 denotes the ring Z/4Z with elements 0,1,2,3. Set R(3) := Ms(Z 4 ) and ^ Then x 2 ± 0, x 3 = 0 = y 2, /0 3 2\ /0 2Z 4 0\ (it+jl)'^ 0 1 3J, and Ann{z 2 } n R(3)y = I 0 2Z 4 Ol. \1 0 0/ \0 2Z 4 0/ REFERENCES [l] P.R. Puchs, 'A characterisation result for matrix rings', Bull. Austral. Math. Soc. 43 (1991),

6 516 N.J. Groenewald and L. van Wyk [6] [2] P.R. Fuchs, C.J. Maxson and G.F. Pilz, 'On rings for which homogeneous maps are linear', PTOC. Amer. Math. Soc. 112 (1991), 1-7. [3] J.C. Robson, 'Recognition of matrix rings', Comm. Algebra 19 (1991), Department of Mathematics University of Port Elizabeth PO Box Port Elizabeth South Africa Department of Mathematics University of Stellenbosch Private Bag X Stellenbosch South Africa lvwqmaties.sun.ac.za